R^{p}t^{q}µ u^{p}t, x^{q}^{}F_{γ}u^{p}t, x^{q}. (4.12)
In this case, Theorem 4.2.1 allows to built a particular solution called the Floquet eigenvector, starting
from the Perron eigenvector which correspond to constant parameters. Moreover, we can compare the
associated Floquet eigenvalue to the Perron eigenvalue. The results about this problem are stated in
Section 4.6.

Before treating the different examples, we explain in Section 4.2 the general method used to tackle these problems. It is based on the main result in Theorem 4.2.1 and the use of the eigenelements of the growth-fragmentation operator together with General Relative Entropy techniques. In Sections 4.3, 4.4, 4.5 and 4.6, we give the proofs of the results in Examples 1, 2, 3 and 4 respectively.

### 4.2 Technical Tools and General Method

### 4.2.1 Main Theorem

The proofs of the main theorems of this paper are based on the following result which requires to consider
that τ^{p}x^{q}is linear (i.e. ν^{}1^{q}.In this case, there exists a relation between a solution to Equation (4.1)
with time-dependent parameters ^{p}V1^{p}t^{q}, R1^{p}t^{qq} and a solution to the same equation with parameters

pV2^{p}t^{q}, R2^{p}t^{qq}. More precisely, we can obtain one from the other by an appropriate dilation. The
fol-lowing theorem generalizes the change of variable used in [47] to build self-similar solution to the pure
fragmentation equation.

Theorem 4.2.1. Consider that Assumptions (4.4)-(4.7)are satisfied with ν^{}1 andγ^{¡}0.For u1^{p}t, x^{q}
a solution to Equation (4.1)with parameters ^{p}V1, R1^{q}, the function u2^{p}t, x^{q}defined by

u2^{p}t, x^{q}^{}W^{}^{k}^{p}t^{q}u1 h^{p}t^{q}, W^{}^{k}^{p}t^{q}x

e^{µ}^{´}^{0}^{t}^{p}^{W}^{p}^{s}^{q}^{R}^{1}^{p}^{s}^{q}^{R}^{2}^{p}^{s}^{qq}^{ds} (4.13)
with k^{}_{γ}^{1}, is a solution to Equation (4.1)with^{p}V2, R2^{q}if W ^{¡}0 andhsatisfy

W9 ^{} τ W

k ^{p}V2^{}V1W^{q}, (4.14)

h9 ^{} W.

Conversely, ifh:R ÑR is one to one and ifu2is a solution with ^{p}V2, R2^{q}, thenu1defined by (4.13)
is a solution with^{p}V1, R1^{q}.

The proof of this result is nothing but easy calculation that we leave to the reader.

Remark 4.2.2. To check that h is one to one, we can take advantage that ODE (4.14)is a Bernoulli equation which can be integrated in

W^{p}t^{q}^{} W0e^{τ}^{k}^{´}^{0}^{t}^{V}^{2}^{p}^{s}^{q}^{ds}
1 W0τ

k

´t

0V1^{p}s^{q}e^{τ}^{k}^{´}^{0}^{s}^{V}^{2}^{p}^{s}^{1}^{q}^{ds}^{1}ds. (4.15)
To tackle the different examples, we use Theorem 4.2.1 together with two techniques appropriate for
this type of equations. First we recall the existence of particular solutions to the growth-fragmentation
equation in the case of time-independent coefficients. They correspond to eigenvectors to the
growth-fragmentation operator and we can give their self-similar dependency on parameters in the case of
pow-erlaw coefficients (see [59, 17]). This dependency is the starting point which leads to Theorem 4.2.1 and
it allows to built an invariant manifold for Equation (4.1) in the caseν ^{}1.It also provides interesting
properties on the moments of the solutions whenν ^{}1.Then we recall results about the General Relative
Entropy (GRE) introduced by [98, 99, 110] for the growth-fragmentation model. This method ensures
that the particular solutions built from eigenvectors are attractive for suitable norms.

### 4.2.2 Eigenvectors and Self-similarity: Existence of an Invariant Manifold

When the coefficients of Equation (4.1) do not depend on time, one can build solutions^{p}t, x^{q}^{ÞÑ}Upx^{q}e^{Λt}
by solving the Perron eigenvalue problem

$

&

%

ΛU^{p}x^{q}^{}^{}^{B}

Bx^{p}τ^{p}x^{q}U^{p}x^{qq}^{}µ^{p}x^{q}U^{p}x^{q} ^{p}F U^{qp}x^{q}, x^{¥}0,
τ^{p}0^{q}U^{p}0^{q}^{}0, U^{p}x^{q}^{¥}0, ´^{8}

0 U^{p}x^{q}dx^{}1.

(4.16)

The existence of such elements Λ and U has been first studied by [95, 111] and is proved for general
coefficients in [41]. The dependency of these elements on parameters is of interest to investigate the
existence of steady states for nonlinear problems (see [20, 17]). In the case of powerlaw coefficients, we
can work out this dependency on frozen transport parameterV and death parameterR(see [59]). Under
Assumptions (4.5)-(4.7), the necessary condition which appears in [41, 95] to ensure the existence of
eigenelements isγ 1^{}ν ^{¡}0.Then we define adilation parameter

k:^{} 1

γ 1^{}ν ^{¡}0 (4.17)

and we have explicit self-similar dependencies

Λ^{p}V, R^{q}^{}V^{kγ}Λ^{p}1,0^{q}^{}Rµ and U^{p}V;x^{q}^{}V^{}^{k}U^{p}1;V^{}^{k}x^{q}. (4.18)
The eigenvectorU does not depend onR,that is why we do not label it. Thereafter Λ^{p}1,0^{q}andU^{p}1;^{q}are
denoted by Λ andU for the sake of clarity. The result of Theorem 4.2.1 is based on the idea to use these
dependencies to tackle time-dependent parameters. An intermediate result between the formula (4.13)
and the dependencies (4.18) is given by the following corollary.

Corollary 4.2.3. Under the assumptions of Theorem 4.2.1, ifW is a solution to

W9 ^{} Λ^{p}W,0^{q}

k ^{p}V ^{}W^{q} (4.19)

then

u^{p}t, x^{q}^{}U^{p}W^{p}t^{q};x^{q}e^{´}^{0}^{t}^{Λ}^{p}^{W}^{p}^{s}^{q}^{,R}^{p}^{s}^{qq}^{ds} (4.20)
is a solution to Equation (4.1).

Proof. For ν ^{}1,we can compute Λ^{}Λ^{p}1,0^{q}by integrating Equation (4.16) with µ^{}0 againstx dx.

We obtain thanks to themass preservation ofF Λ

ˆ ^{8}

0

xUpx^{q}dx^{}τ
ˆ ^{8}

0

xUpx^{q}dx

and so Λ^{p}1,0^{q}^{}τ.Thus using the dependency (4.18) we find that Λ^{p}W,0^{q}^{}τ W and Equation (4.19)
is nothing but a rewriting of Equation (4.14). We use this formulation (4.19) here to highlight the link
with eigenelements, and because it allows after to obtain results in the cases whenν^{}1.Now we apply
Theorem 4.2.1 forV1^{}1, R1^{}0 andV2^{}V, R2^{}Rand we obtain that

u2^{p}t, x^{q}^{}W^{}^{k}U^{p}W^{}^{k}x^{q}e^{Λh}^{p}^{t}^{q}^{´}^{0}^{t}^{R}^{p}^{s}^{q}^{ds}^{}U^{p}W^{p}t^{q};x^{q}e^{´}^{0}^{t}^{Λ}^{p}^{W}^{p}^{s}^{q}^{,R}^{p}^{s}^{qq}^{ds}
is a solution to Equation (4.1).

This corollary provides a very intuitive explicit solution in the spirit of dependencies (4.18). At each
time t, the solution is an eigenvector associated to a parameter W^{p}t^{q} with an instantaneous fitness
Λ^{p}W^{p}t^{q}, R^{p}t^{qq}associated to the same parameterW^{p}t^{q}.The function t^{ÞÑ}W^{p}t^{q}thus defined follows V^{p}t^{q}
with a delay explicitely given by ODE (4.19).

A very useful consequence for the different applications is the existence of an invariant manifold for the growth-fragmentation equation with time-dependent parameters of the form (4.4). Let define the eigenmanifold

E:^{} QUpW;^{q}, ^{p}W, Q^{q}^{P}^{p}R^{}q

2^{(} (4.21)

which is the union of all the positive eigenlines associated to a transport parameter W. Then
Corol-lary 4.2.3 ensures that, under the assumptions of Theorem 4.2.1, any solution to Equation (4.1) with an
initial distributionu_{0}^{P}E remains inE for all time. Moreover the dynamics of such a solution reduces to
the ODE (4.19) and this is the key point we use to tackle nonlinear problems.

Forν^{}1,the technique fails and we cannot obtain explicit solution with the method of Theorem 4.2.1.

Nevertheless, we can still define W as the solution to ODE (4.19) and give properties of the functions
defined as dilations of the eigenvector by (4.20). We obtain that the moments of these functions satisfy
equations that are similar to the ones verified by the moments of the solution to the growth-fragmentation
equation. In the special caseν ^{}0,andγ^{}1,we even obtain the same equation. More precisely, if we
denote, forα^{¥}0,

M_{α}^{r}W^{sp}t^{q}:^{}
ˆ ^{8}

0

x^{α}U^{p}W^{p}t^{q};x^{q}e^{´}^{0}^{t}^{Λ}^{p}^{W}^{p}^{s}^{q}^{,R}^{p}^{s}^{qq}^{ds}dx, (4.22)
and

Mα^{r}u^{sp}t^{q}:^{}
ˆ ^{8}

0

x^{α}u^{p}t, x^{q}dx, (4.23)

then we have the following result.

Lemma 4.2.4. On the one hand, if W is a solution to Equation (4.19), then the moments M_{α}satisfy
M9_{α}^{}αΛaαVM_{α ν}1 ^{p}1^{}α^{q}ΛbαM_{α γ}^{}µRM_{α} (4.24)
with

a_{α}:^{} M_{α}^{r}Us

M_{α ν}1^{r}U^{s} and b_{α}:^{} M_{α}^{r}Us

M_{α γ}^{r}U^{s}.

On the other hand, ifuis a solution to the fragmentation-drift equation, then the momentsMα satisfy
M9α^{}ατ V Mα ν^{}1 ^{p}cα^{}1^{q}βMα γ^{}µRMα (4.25)
with

c_{α}:^{}
ˆ 1

0

z^{α}κ^{p}z^{q}dz.

Proof. Thanks to a change of variable we can compute for allα^{¥}0
M_{α}^{r}W^{s}^{}Mα^{r}U^{s}W^{kα}e^{´}^{0}^{t}^{Λ}^{p}^{W}^{p}^{s}^{q}^{,R}^{p}^{s}^{qq}^{ds}
so we have

M9_{α} kα
W9

WM_{α} Λ^{p}W, R^{q}M_{α}

αΛW^{kγ}^{}^{1}^{p}V ^{}W^{q}M_{α} ΛW^{kγ}M_{α}^{}µRM_{α}

αΛV W^{k}^{p}^{ν}^{}^{1}^{q}M_{α} p1^{}α^{q}ΛW^{kγ}M_{α}µRM_{α}

αΛV aαM_{α ν}1 ^{p}1^{}α^{q}ΛbαM_{α γ}^{}µRM_{α}.

Integrating Equation (4.1) againstx^{α}dx we obtain by integration by part and thanks to the Fubini

theorem
α^{}1 orα^{}2.A consequence is the useful result given in the following corollary.

Corollary 4.2.5.In the case whenν ^{}0, γ^{}1andκsymmetric, the zero and first moments^{p}M_{0},M_{1}q

This Corollary allows in Section 4.6 to compare the Perron and Floquet eigenvalues not only forν ^{}1
but also forν^{}0, γ^{}1.We do not have in this case a particular solution to the growth-fragmentation
equation as in Corollary 4.2.3, but a particular solution of the reduced ODE system (4.26).

Proof. Forκsymmetric, we have already seen thatc0^{}n0^{}2.Together with Assumption (4.3) which
givesc1^{}1,we conclude that^{p}M0, M1^{q}is solution to Equation (4.26).

Integrating Equation (4.16) againstdxandx dx we obtain
Λ^{}β

### 4.2.3 General Relative Entropy: Attractivity of the Invariant Manifold

The existence of the invariant manifold E is useful to obtain particular solutions to nonlinear growth-fragmentation equations. But what happens when the initial distribution u0 does not belong to this manifold ? The GRE method ensures thatE is attractive in a sense to be defined.

The GRE method requires to consider the adjoint growth-fragmentation equation

B

Btψ^{p}t, x^{q}^{}τ^{p}t, x^{q} ^{B}

Bxψ^{p}t, x^{q}^{}µ^{p}t, x^{q}ψ^{p}t, x^{q} ^{p}F^{}ψ^{qp}t, x^{q} (4.27)
whereF^{} is the adjoint fragmentation operator

pF^{}ψ^{qp}t, x^{q}:^{}
ˆ x

0

b^{p}t, x, y^{q}ψ^{p}t, y^{q}dy^{}β^{p}t, x^{q}ψ^{p}t, x^{q}.

Ifuand v are two solutions to Equation (4.1) andψ is a solution to Equation (4.27), then we have for

WhenHis convex, the right hand side is nonpositive and we obtain a nonincreasing quantity called GRE.

In the case of time-independent coefficients, we can chose forv a solution of the formU^{p}x^{q}e^{Λt}.Then,
to apply the GRE method, we need a solution to the adjoint equation and such solutions are given by
solving the adjoint Perron eigenvalue problem

$

Such a problem is usually solved together with the direct problem (4.16) and the first eigenvalue Λ is
the same for the two ones (see [41, 95, 110]). Then the GRE ensures that any solutionuto the
growth-fragmentation equation behaves asymptotically likeU^{p}x^{q}e^{Λt}.More precisely it is proved in [99, 110] under
general assumptions that

tlim^{Ñ8}

}̺^{}_{0}^{1}u^{p}t,^{q}e^{}^{Λt}^{}U}L^{p}^{p}U^{1}^{}^{p}φ dx^{q}^{}0, ^{}p^{¥}1. (4.29)
where̺0^{}´

u0^{p}y^{q}φ^{p}y^{q}dy withu0^{p}x^{q}^{}u^{p}t^{}0, x^{q}.

Now consider Equation (4.1) whose coefficients are time-dependent. Under the assumptions of
The-orem 4.2.1 and for p ^{} 1, the convergence result (4.29) can be interpreted as the attractivity of the
invariant manifoldE inL^{1}^{p}φ dx^{q}with the distance

d^{p}u,E^{q}:^{} inf
to one and we can define from a solutionu^{p}t, x^{q}to Equation (4.1) the function

v^{p}h^{p}t^{q}, x^{q}:^{}W^{k}^{p}t^{q}u^{p}t, W^{k}^{p}t^{q}x^{q}e^{}^{´}^{0}^{t}^{p}^{W}^{p}^{s}^{q}^{µR}^{p}^{s}^{qq}^{ds} (4.30)
so, denoting byU and φthe eigenfunctions of Equation (4.31), we have

ˆ ^{8}

Forν ^{}1, φis linear (see examples in [41]), so we can compute from (4.30)

̺^{p}t^{q}^{}
ˆ

u^{p}t, y^{q}φ^{p}y^{q}dy^{}̺0W^{k}^{p}t^{q}e^{´}^{0}^{t}^{p}^{W}^{p}^{s}^{q}^{µR}^{p}^{s}^{qq}^{ds}
and

d^{p}u^{p}t,^{q},E^{q}^{¤}^{}}̺^{}^{1}^{p}t^{q}u^{p}t,^{q}^{}W^{}^{k}U^{p}W;^{q}}^{}^{}}̺^{}_{0}^{1}v^{p}h^{p}t^{q},^{q}^{}U^{}}^{Ñ}0. (4.32)
The exponential decay in (4.29) is proved in [111, 79] forp^{}1 and for a constant fragmentation rate
β^{p}x^{q}^{}β. It is also proved in [15] for powerlaw parameters in the norm corresponding top^{}2 and this
is the case we are interested in. Assume that the coefficients satisfy (4.5), (4.6) and (4.7) and assume
also that the fragmentation kernel is upper and lower bounded

Dκ, κ^{¡}0, ^{}z^{P}^{r}0,1^{s}, κ^{¤}κ^{p}z^{q}^{¤}κ. (4.33)
Then a spectral gap result is proved in [15] forν ^{}1 inL^{2}^{p}U^{}^{1}φ dx^{q}and then the result is extended to
bigger spaces thanks to a general method for spectral gaps in Hilbert spaces.

Theorem [15]. Under Assumption (4.5) with ν ^{} 1, Assumptions (4.6) and (4.33), and
Assump-tion (4.7)with γ ^{P}^{p}0,2^{s}, there exista¯^{¡}0 andr¯^{¥}3 such that, for any a^{P}^{p}0,¯a^{q}and anyr^{¥}r,¯ there
existsCa,r such that for anyu0^{P}H:^{}L^{2}^{p}θ^{q}, θ^{p}x^{q}^{}x x^{r},there holds

t^{¡}0, ^{}}̺^{}_{0}^{1}u^{p}t,^{q}e^{}^{Λt}^{}U}H ^{¤}C_{a,r}^{}}̺^{}_{0}^{1}u0^{}U}He^{}^{at}. (4.34)
This result is very useful for Applications 1 and 2 becauseL^{2}^{p}θ^{q}^{}L^{1}^{p}x^{p}^{q} forr ^{¥}2p 1. Moreover
the exponential decay allows to prove exponential stability results for Equation (4.8) whenκis constant
(see Section 4.3).